Analyzing Profit Functions

Question

For a certain company, the cost function for producing x items is C(x)=50x+150 and the revenue function for selling x items is R(x)=−0.5(x−110)2+6,050. The maximum capacity of the company is 170 items.

1. Assuming that the company sells all that it produces, what is the profit function?
2. What is the domain of P(x)?
3. The company can choose to produce either 60 or 70 items. What is their profit for each case, and which level of production should they choose?

Profit when producing 60 items =    

Profit when producing 70 items =

1. Can you explain, from our model, why the company makes less profit when producing 10 more units?

Really having trouble with this one any help would be great.

Answer 1

1. Assuming that the company sells all that it produces, what is the profit function?

Profit is the difference between revenue and total cost i.e., P(x)=R(x)-C(x). In this question,

P(x)=[-0.5(x-110)2+6,050]-[50x+150] which simplifies to P(x)=-51x+6,010.

2. What is the domain of P(x)?

Since x represents the number of items produced and sold, it cannot be a negative number. In addition, we are not given the maximum number of items the company can produce and sell. Hence, the domain of P(x) must be all non-negative numbers i.e., x≥0 OR [0,∞).

3. The company can choose to produce either 60 or 70 items. What is their profit for each case, and which level of production should they choose?

Profit when producing 60 items = P(x=60)=-51(60)+6,010=$2950.

Profit when producing 70 items = P(x=70)=-51(70)+6,010=$2440.

From the above calculations, the company should choose a production level of 60 items as this production level yields a bigger profit compared to a production level of 70 items.

1. Can you explain, from our model, why the company makes less profit when producing 10 more units?

The model [i.e., the profit function P(x)=-51x+6,010] is a linear function with a negative slope. In other words, it is a decreasing function which means that the company makes less and less profit with each additional item that is produced and sold. This explains why the company makes less profit when producing 10 more units i.e., when production is increased from 60 items to 70 items.